A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups

TL;DR

This paper establishes a Tukia-type theorem for nilpotent Lie groups, demonstrating conditions under which uniform quasiconformal groups can be conjugated into conformal groups, with implications for the quasi-isometric rigidity of solvable groups.

Contribution

It extends Tukia-type theorems to nilpotent Lie groups and applies these results to prove quasi-isometric rigidity of specific solvable groups.

Findings

Uniform quasiconformal groups can be conjugated into conformal groups under cocompact action.

Results apply to Carnot-by-Carnot groups and their quasi-isometric rigidity.

Provides new tools for understanding the geometric structure of solvable groups.

Abstract

In this paper we study uniform quasiconformal groups of Carnot-by-Carnot groups. We show that they can be conjugated into conformal groups provided the induced action on the space of distinct pairs is cocompact. Following the approach of Eskin-Fisher-Whyte these results have applications to quasi-isometric rigidity of certain solvable groups.